INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS
11 CFD ACTIVITY 12 TRANSPORT EQUATIONS 13 NUMERICAL VERSUS ANALYTICAL SOLUTIONS 14 MAIN TASK 15 A NOTE ON NAVIER–STOKES EQUATIONS 16 OUTLINE OF THE BOOK EXERCISES 21 INTRODUCTION 22 1D CONDUCTION EQUATION 23 GRID LAYOUT 24 DISCRETISATION 241 TSE METHOD 242 IOCV METHOD 25 STABILITY AND CONVERGENCE 251 EXPLICIT PROCEDURE ? = 252 PARTIALLY IMPLICIT PROCEDURE 0 < ? < 253 IMPLICIT PROCEDURE ? = 26 MAKING CHOICES 27 DEALING WITH NONLINEARITIES 271 NONLINEAR SOURCES 272 NONLINEAR COEF?CIENTS 273 BOUNDARY CONDITIONS 274 UNDERRELAXATION 28 METHODS OF SOLUTION 281 GAUSS–SEIDEL METHOD 282 TRIDIAGONAL MATRIX ALGORITHM 283 APPLICATIONS 29 PROBLEMS FROM RELATED FIELDS 3 1D CONDUCTION–CONVECTION 31 INTRODUCTION 32 EXACT SOLUTION 33 DISCRETISATION 34 UPWIND DIFFERENCE SCHEME 35 COMPARISON OF CDS, UDS, AND EXACT SOLUTION 36 NUMERICAL FALSE DIFFUSION 37 HYBRID AND POWER-LAW SCHEMES 38 TOTAL VARIATION DIMINISHING SCHEME 39 STABILITY OF THE UNSTEADY EQUATION 391 EXACT SOLUTION 392 EXPLICIT FINITE-DIFFERENCE FORM 393 IMPLICIT FINITE-DIFFERENCE FORM 41 GOVERNING EQUATIONS 42 ADAPTIVE GRID 43 TRANSFORMATION TO (X, ?) COORDINATES 44 DISCRETISATION 45 DETERMINATION OF ?, Y, AND R 46 BOUNDARY CONDITIONS 461 SYMMETRY 462 WALL 463 FREE STREAM 47 SOURCE TERMS 471 PRESSURE GRADIENT 472 Q??? AND RK 48 TREATMENT OF TURBULENT FLOWS 481 MIXING LENGTH MODEL 482 E?? MODEL 483 FREE-SHEAR FLOWS OVERALL PROCEDURE 491 CALCULATION SEQUENCE 492 INITIAL CONDITIONS 493 CHOICE OF STEP SIZE AND ITERATIONS 410 APPLICATIONS 5 2D CONVECTION – CARTESIAN GRIDS 51 INTRODUCTION MAIN TASK SOLUTION STRATEGY 52 SIMPLE – COLLOCATED GRIDS MAIN IDEA DISCRETISATION PRESSURE-CORRECTION EQUATION FURTHER SIMPLI?CATION OVERALL CALCULATION PROCEDURE 53 METHOD OF SOLUTION ITERATIVE SOLVERS EVALUATION OF RESIDUALS UNDERRELAXATION BOUNDARY CONDITIONS FOR BOUNDARY CONDITION FOR P? M NODE TAGGING 54 TREATMENT OF TURBULENT FLOWS LRE MODEL HRE MODEL 55 NOTION OF SMOOTHING PRESSURE CORRECTION 56 APPLICATIONS 6 2D CONVECTION – COMPLEX DOMAINS 61 INTRODUCTION CURVILINEAR GRIDS UNSTRUCTURED GRIDS 62 CURVILINEAR GRIDS COORDINATE TRANSFORMATION TRANSPORT EQUATION INTERPRETATION OF TERMS NODE TAGGING AND BOUNDARY CONDITIONS 63 UNSTRUCTURED MESHES GAUSS’S DIVERGENCE THEOREM CONSTRUCTION OF A LINE STRUCTURE CONVECTIVE TRANSPORT DIFFUSION TRANSPORT INTERIM DISCRETISED EQUATION INTERPOLATION OF AT P2, E2, A, AND B FINAL DISCRETISED EQUATION EVALUATION OF NODAL GRADIENTS 6310 BOUNDARY CONDITIONS 6311 PRESSURE-CORRECTION EQUATION 6312 METHOD OF SOLUTION 6313 OVERALL CALCULATION PROCEDURE 64 APPLICATIONS 65 CLOSURE 71 INTRODUCTION 72 1D PROBLEMS FOR PURE SUBSTANCES EXACT SOLUTION SIMPLE NUMERICAL SOLUTION NUMERICAL SOLUTION USING TDMA ACCURATE SOLUTIONS ON A COARSE GRID 73 1D PROBLEMS FOR IMPURE SUBSTANCES 8 NUMERICAL GRID GENERATION 81 INTRODUCTION 82 ALGEBRAIC GRID GENERATION 1D DOMAINS 2D DOMAINS 83 DIFFERENTIAL GRID GENERATION INVERSION OF DETERMINANT EQUATIONS 84 SORENSON’S METHOD MAIN SPECI?CATIONS STRETCHING FUNCTIONS 843 DISCRETISATION 844 SOLUTION PROCEDURE 845 APPLICATIONS 85 UNSTRUCTURED MESH GENERATION 851 MAIN TASK 852 DOMAINS WITH (I , J ) STRUCTURE 853 AUTOMATIC GRID GENERATION 9 CONVERGENCE ENHANCEMENT 91 CONVERGENCE RATE 92 BLOCK CORRECTION 93 METHOD OF TWO LINES 94 STONE’S METHOD 95 APPLICATIONS